Vol. 34, No. 2, 1970

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Finite transformation formulae involving the Legendre symbol

Kenneth S. Williams

Vol. 34 (1970), No. 2, 559–568
Abstract

Let p denote an odd prime. The following three identities (transformation formulae) involving the Legendre symbol (-
p) are known to be valid for any complex-valued function F defined on the integers, which is periodic with period p:

x=0p1F(x) + x=0p1(x-
p)F(x) = x=0p1F(x2),
x=0p1F(x) + x=0p1(x2 − 4a
---p---)F(x) = x=1p1F(x + a
x-),a0( mod p),
x=0p1F(x) + x=0p1( 2
x--− 4x
p)F(x) = x=1p1F(x + 2 + 1-
x).
We consider a general class of transformation formulae, which includes the above examples.

Mathematical Subject Classification
Primary: 10.43
Milestones
Received: 26 August 1969
Revised: 30 January 1970
Published: 1 August 1970
Authors
Kenneth S. Williams